Properties of root subspaces
Let be a square matrix and let be its eigenvalue. As we know, the nonzero elements of the null space are the corresponding eigenvectors. This definition is generalized as follows.
Definition 1. The subspaces are called root subspaces of corresponding to
Exercise 1. a) Root subspaces are increasing:
(1) for all
and b) there is such that all inclusions (1) are strict for and
Proof. a) If for some then which shows that
b) (1) implies Since all root subspaces are contained in there are such that Let be the smallest such Then all inclusions (1) are strict for
Suppose for some Then there exists such that , that is, Put Then This means
that which contradicts the definition of
Definition 2. Property (2) can be called stabilization. The number from (2) is called a height of the eigenvalue .
Exercise 2. Let and let be the number from Exercise 1. Then
Proof. By the rank-nullity theorem applied to we have By Exercise 3, to prove (3) it is sufficient to establish that Let's assume that contains a nonzero vector Then we have for some We obtain two facts:
It follows that is a nonzero element of This contradicts (2). Hence, the assumption is wrong, and (3) follows.
Exercise 3. Both subspaces at the right of (3) are invariant with respect to
Proof. If then by commutativity of and we have so
Suppose so that for some Then
Exercise 3 means that, for the purpose of further analyzing we can consider its restrictions onto and
Exercise 4. The restriction of onto does not have eigenvalues other than
Proof. Suppose for some Since we have Then and . This implies
Exercise 5. The restriction of onto does not have as an eigenvalue (so that is invertible).
Proof. Suppose and Then for some and By Exercise 1 and This contradicts the choice of